Common Knowledge, Regained

TL;DR

This paper introduces a new, more permissive definition of common knowledge that can arise without simultaneity in dynamic settings, resolving a longstanding paradox and enabling new results in game theory.

Contribution

It proposes a novel definition of common knowledge applicable in dynamic settings, resolving the Halpern-Moses paradox and extending theoretical results.

Findings

Common knowledge can arise without simultaneity under the new definition.

The new definition enables an agreement theorem in dynamic settings with timing frictions.

Application to a dynamic coordination game characterizes equilibrium behavior.

Abstract

For common knowledge to arise in dynamic settings, all players must simultaneously come to know it has arisen. Consequently, common knowledge cannot arise in many realistic settings with timing frictions. This counterintuitive observation of Halpern and Moses (1990) was discussed by Arrow et al. (1987) and Aumann (1989), was called a paradox by Morris (2014), and has evaded satisfactory resolution for four decades. We resolve this paradox by proposing a new definition for common knowledge, which coincides with the traditional one in static settings but is more permissive in dynamic settings. Under our definition, common knowledge can arise without simultaneity, particularly in canonical examples of the Haplern-Moses paradox. We demonstrate its usefulness by deriving for it an agreement theorem \`a la Aumann (1976), showing it arises in the setting of Geanakoplos and Polemarchakis (1982) with timing frictions added, and applying it to characterize equilibrium behavior in a dynamic coordination game.